Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations

TL;DR

This paper derives exact energy levels and wavefunctions for a harmonic oscillator with modified commutation relations inspired by string theory, revealing potential observable effects in high-precision experiments.

Contribution

It provides an exact solution for the harmonic oscillator under minimal length uncertainty relations, connecting string theory concepts with quantum mechanics.

Findings

Exact eigenvalues and eigenfunctions obtained

Modified commutation relations influence quantum states

Potential experimental signatures in high-precision measurements

Abstract

We determine the energy eigenvalues and eigenfunctions of the harmonic oscillator where the coordinates and momenta are assumed to obey the modified commutation relations [x_i,p_j]=i hbar[(1+ beta p^2) delta_{ij} + beta' p_i p_j]. These commutation relations are motivated by the fact they lead to the minimal length uncertainty relations which appear in perturbative string theory. Our solutions illustrate how certain features of string theory may manifest themselves in simple quantum mechanical systems through the modification of the canonical commutation relations. We discuss whether such effects are observable in precision measurements on electrons trapped in strong magnetic fields.